Sure, let's work through this step by step for each part of the question.
### (a) Compute [tex]\( f(11) \)[/tex]
The probability mass function for a binomial distribution is given by:
[tex]\[ f(k) = P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \][/tex]
For [tex]\( n = 20 \)[/tex], [tex]\( p = 0.70 \)[/tex], and [tex]\( k = 11 \)[/tex]:
[tex]\[ f(11) \approx 0.0654 \][/tex]
### (b) Compute [tex]\( f(16) \)[/tex]
Using the same probability mass function for a binomial distribution:
For [tex]\( n = 20 \)[/tex], [tex]\( p = 0.70 \)[/tex], and [tex]\( k = 16 \)[/tex]:
[tex]\[ f(16) \approx 0.1304 \][/tex]
### (c) Compute [tex]\( P(x \geq 16) \)[/tex]
The probability of [tex]\( x \)[/tex] being greater than or equal to 16 is the sum of individual probabilities from 16 to 20. This is often easier calculated using the survival function:
[tex]\[ P(x \geq 16) \approx 0.2375 \][/tex]
### (d) Compute [tex]\( P(x \leq 15) \)[/tex]
The probability of [tex]\( x \)[/tex] being less than or equal to 15 is the cumulative distribution function (CDF) evaluated at 15:
[tex]\[ P(x \leq 15) \approx 0.7625 \][/tex]
### (e) Compute [tex]\( E(x) \)[/tex]
The expected value [tex]\( E(x) \)[/tex] for a binomial distribution is given by:
[tex]\[ E(x) = n \cdot p \][/tex]
For [tex]\( n = 20 \)[/tex] and [tex]\( p = 0.70 \)[/tex]:
[tex]\[ E(x) = 20 \cdot 0.70 = 14.0 \][/tex]
### (f) Compute [tex]\( \operatorname{Var}(x) \)[/tex] and [tex]\( \sigma \)[/tex]
The variance [tex]\( \operatorname{Var}(x) \)[/tex] for a binomial distribution is:
[tex]\[ \operatorname{Var}(x) = n \cdot p \cdot (1 - p) \][/tex]
For [tex]\( n = 20 \)[/tex] and [tex]\( p = 0.70 \)[/tex]:
[tex]\[ \operatorname{Var}(x) = 20 \cdot 0.70 \cdot 0.30 = 4.2000 \][/tex]
The standard deviation [tex]\( \sigma \)[/tex] is the square root of the variance:
[tex]\[ \sigma = \sqrt{\operatorname{Var}(x)} = \sqrt{4.2000} \approx 2.0494 \][/tex]
### Summary:
- [tex]\( f(11) \approx 0.0654 \)[/tex]
- [tex]\( f(16) \approx 0.1304 \)[/tex]
- [tex]\( P(x \geq 16) \approx 0.2375 \)[/tex]
- [tex]\( P(x \leq 15) \approx 0.7625 \)[/tex]
- [tex]\( E(x) = 14.0 \)[/tex]
- [tex]\( \operatorname{Var}(x) = 4.2000 \)[/tex]
- [tex]\( \sigma \approx 2.0494 \)[/tex]
